ANTENNA GAIN-FACTOR EQUIVALENT FOR TEM CELLS Perry F. Wilson National Institute of Standards and Technology 325 Broadway, Boulder, CO 80305, USA Email: [email protected] Abstract: This paper derives a gain-factor equivalent where E is the peak electric-field amplitude and η0 is for TEM cells. The gain factor is then used in simple the free-space wave impedance. Neglecting non-TEM transmission formulas to investigate emissions tests in modes, the peak electric field amplitude near the TEM cells and correlation of TEM-cell emissions center of the test volume in a TEM cell is measurements to other methods. The correlation approximately the voltage amplitude V between the equations are not restricted to electrically small test inner and outer conductor divided by their separation objects, as is the case for dipole models. Thus, these h, or E ≈ V / h . Thus, results are of use as EMC test methods, and standards are extended to frequencies exceeding 1 GHz. 2 1 1 V S av,TEM ≈ . (3) I. INTRODUCTION 2 η0 h The equipment under test (EUT) in a TEM cell is The average input power to a TEM cell is related to typically modeled as a set of dipoles when one the input voltage and characteristic impedance Z0 of analyzes emission measurements [1-2]. Dipole models the cell: work well for electrically small emitters but may not accurately describe emissions from electrically large 1 V 2 emitters. Thus, a more general emission model is Pt = . (4) 2 Z needed. One approach is to account for further terms 0 (e.g., quadrapole) in a multipole model of the EUT [3- 4]; however, higher-order multipole models quickly Combining these results yields become cumbersome and may not lead to simple, usable measurement procedures. This paper develops Z 0 1 S ≈ P (5) av,TEM t 2 a simple alternative for the general emissions case, η0 h namely, an antenna gain-factor equivalent for TEM cells. Having a gain factor allows TEM cells to be for the average power density in a TEM cell. This is used for emissions testing in the same manner as similar in form to the free-field expression (2) and can unguided wave test environments, such as fully be reformulated in equivalent parameters. For the case anechoic rooms (FAR), semi-anechoic room (SAR), of a TEM cell with a constant flare of angle θ in the and open area test sites (OATS). test volume (e.g., a gigahertz TEM (GTEM) cell where typicallyθ ≈ 15° ), h ≈ r sinθ , where r is the II. TEM CELL IN TRANSMITTING MODE distance from the apex (input/output port) to the receiving antenna location. For a TEM cell with In free space, the far-field (subscript FF) average uniform test section (tapers at each end), h ≈ r sinθ , power density Sav,FF at a distance r from a source is where r is the distance from the apex to the receiving antenna location projected back along the uniform 1 S = P g , (1) section to the end of the taper. Substituting this av,FF t t 2 4πr approximation and rewriting yields where Pt and gt are the average power and gain of the 4πZ 1 source (transmitter). S ≈ P 0 . (6) av,TEM t 2 2 η0 sin θ 4πr The TEM mode in a TEM cell approximates an ideal plane wave (uniform, linearly polarized) over a This expression is for the average power density near limited test volume. Thus, the average power density the center of the test volume in a TEM cell. Sav,TEM for the TEM mode is Comparing equations (1) and (6) we see that 2 1 E π S = , (2) 4 Z 0 av,TEM g t,TEM ≈ (7) 2 η0 2 η0 sin θ defines an equivalent gain factor for the TEM cell. where k0 is the wavenumber, e0y is a normalized field Alternately, gt,TEM can be derived directly from the 2 2 factor ( e ≈ Z / h , [2]), and SV is related to definition of gain, 0 y 0 2 2 measured output port voltage ( SV ≈ V for the S g = 4πr 2 r , (8) electric dipole as emitter [2]; a subscript V has been Pt added here to avoid confusion with power density). by substituting (3) for the power density in the Substituting these results into (12) yields direction of the test volume, (4) for the transmitted 2 2 power, and again using the approximation h ≈ r sinθ . η0 k0 h 2 P0 = V . (13) The advantage of the first derivation is that it 3π Z 2 emphasizes the geometrical similarities between a 0 TEM cell and an antenna. 2 Substituting in V from (4), with h ≈ r sinθ and k0 = The gain factor can now be used in transmission 2π/λ, gives formulas to consider simple coupling problems. In 2 2 2 particular, we will use the Friis formula, η0 2π r sin θ P0 = 2Z 0 Pr . (14) 3π λ 2 2 Z 0 λ Pr = Pt gt g r , (9) 4πr Solving for Pr yields 2 that relates received Pr and transmitted Pt power π 3 4 Z 0 λ between a receiving antenna with gain gr and a Pr = P0 , (15) 2 η sin 2 θ 4πr transmitting antenna with gain gt. 0 III. TEM CELL IN RECEIVING MODE which is the same as (11). Thus, (11) is consistent with previously derived dipole equations. For a general antenna (EUT) in the transmitting mode, the received power at the output port of a TEM cell III.2 Electrically Large EUT may now be written from eqs. (7) and (9) as For electrically large unintentional emitters, the 2 4πZ λ expected value (< >) of the directivity (maximum P = P g 0 . (10) gain) can be estimated as [5] r t t 2 π η0 sin θ 4 r This expression is next applied to some specific 1 1 examples. D ≈ 0.577 + ln N s + , (16) 2 2N s III.1 Short Electric Dipole 2 where N s = 4(k0 a) + 8(k0 a) , and a is the radius of Consider a short electric dipole with Pt = P0 and gt = the minimum sphere enclosing the EUT. Electrically 3/2 (maximum gain). By use of (10), the received large is here defined by k0a > 1. power at the output of a TEM cell, for the case of a dipole oriented for maximum coupling, is given by Inserting this expression in (10) yields an expected value for the received power, when the EUT is 2 3 4πZ λ oriented for maximum emissions, given by = 0 Pr P0 . (11) 2 η sin 2 θ 4πr 0 2 4πZ λ P = P D 0 . (17) r t 2 π This result may be compared to previously derived η0 sin θ 4 r expressions based on a waveguide analysis. For example, in IEC 61000-4-20 [2], P0 for an electric dipole is given as This can be used as a means of estimating Pt if the orientation for maximum coupling is known. 2 η k Alternately, Pt can be estimated by averaging over P = 0 0 S 2 , (12) 0 π 2 V multiple orientations [6]. 3 e0 y Z 0 IV. CORRELATION OF MAXIMUM types is difficult due to site imperfections, set-up EMISSIONS BETWEEN VARIOUS TEST SITES variability, operator procedures, and other factors. There is the unfortunate tendency when comparing Correlation between test sites can be used to set data between EMC emission sites to assume that one equivalent test limits and to compare test data. The site yields “correct” results and correlated data from transmission formulas above allow us to write simple other sites must match this “reference data” or be expressions for the received power at various test sites judged to be in “error.” Differences may be caused when the EUT is oriented for maximum emissions. more by each site type accentuating differing aspects We have: of the overall EUT emissions. 2 P 4πZ λ A simple example of this is the spatial sampling used r ≈ 0 (TEM cell), at various sites: planar cuts are typical for a FAR, η π Pt gt 0 4 h sectoral wedges typical for an OATS, and three-axis 2 P λ rotations are typical for a TEM cell. The true r = g (FAR, free space), (18) maximum emissions may or may not be detected by r π Pt g t 4 r any one of these differing schemes. Correlated results 2 may differ simply because the full 3D emission Pr λ ≈ g r 4 (OATS, SAR, ground plane), pattern is sampled in very different ways and not P g 4πr t t because data or correlation algorithms are “in error.” P r ≈ 1 (reverberation chamber). Despite these reservations, an example of measured Pt gt data is presented next. In the above OATS expression, we make the V. MEASURED DATA approximation that the ideal ground plane (infinite size, perfectly conducting) doubles the electric field As part of a site comparison, we obtained data for a (quadruple received power) compared to the free- self-contained comb generator with an attached loop space case when the EUT is oriented for maximum antenna, 30 cm square, as described in [7]. The emissions and constructive interference (via a height overall diameter of the minimum sphere containing scan). In the expression for the reverberation both the loop and comb generator box is chamber, we make the approximation that averaging approximately 60 cm. Thus, a ≈ 0.3 m, and the over multiple paddle positions negates transmitter emitter becomes electrically large (k0a > 1) above 160 gain and only the total transmitted power is measured.